Todaro Smith Economic Development 12th Edition Problem Set

Todaro Smith Economic Development 12th Editionproblem Set Chapte

Todaro & Smith – Economic Development 12th edition Problem set – Chapter 5 1. The following income distribution data are for Brazil. Quintile Percent Share Lowest 20% 3.0% Second quintile 6.9% Third quintile 11.8% Fourth quintile 19.6% Highest 20% 58.7% Highest 10% 43.0% (a) Carefully graph the Lorenz curve, labeling the axes. (b) Explain how to find the Gini coefficient, graphically. (c) Brazil’s national income is about $300 billion. What is the approximate dollar income of the bottom 20%? Bottom 40%? (d) Brazil’s population is approximately 150 million. Suppose that each household makes the average income for its quintile. What is the level of poverty if the poverty line is $400 per capita? (e) Suppose one percent of national income were transferred from the richest 20% of households to the poorest 20% of households. Show the effect on relative inequality. Under the same transfer, what is the effect on poverty?

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Economic inequality is a critical issue in development economics, significantly impacting social cohesion, economic stability, and growth. The problem set based on Brazil’s income distribution provides an insightful framework for understanding income inequality through the Lorenz curve, Gini coefficient, poverty measurement, and the effects of income redistribution. This essay explores these concepts systematically, employing the given data to analyze income disparity, measure inequality, and examine potential policy implications.

Understanding the Lorenz Curve

The Lorenz curve visually represents income distribution within a country by plotting the cumulative percentage of total income earned against the cumulative percentage of the population, ordered from the poorest to the richest. For Brazil, the data indicates that the lowest 20% of the population earn only 3.0% of the total income, while the highest 20% earn 58.7%. The curve begins at (0,0) and, based on the data, will be convex, illustrating the concentration of income among the wealthiest.

Graphically, the axes are labeled as follows: the x-axis represents the cumulative percentage of the population, ranging from 0% to 100%, while the y-axis shows the cumulative percentage of income, also from 0% to 100%. The 45-degree line (line of equality) indicates perfect income equality, where each percentage of the population earns an equal share of income. The Lorenz curve lies beneath this line, with its degree of bowing reflecting inequality—the more bowed, the higher the inequality.

Calculating the Gini Coefficient

The Gini coefficient quantifies income inequality on a scale from 0 (perfect equality) to 1 (maximum inequality). Graphically, it is derived from the Lorenz curve as the ratio of the area between the line of perfect equality and the Lorenz curve to the total area under the line of perfect equality. Mathematically, it can be approximated through numerical integration or formulae based on income shares and population percentiles.

For the provided income shares, the Gini coefficient can be roughly estimated using the Lorenz curve data points. This involves computing the area between the Lorenz curve and the line of equality, then dividing by the total area under the line of equality (which is 0.5). A higher Gini coefficient indicates greater inequality; for Brazil, studies suggest a Gini coefficient around 0.53, consistent with her income distribution data.

Estimating Income for the Bottom Quintile and Quarter

Brazil’s total income is approximately $300 billion. The bottom 20%, earning 3.0%, have an income of:

$300 billion × 0.03 = $9 billion.

Similarly, the bottom 40%, comprising the first two quintiles, earn a combined 3.0% + 6.9% = 9.9% of total income, amounting to:

$300 billion × 0.099 = $29.7 billion.

Thus, the bottom 20% have about $9 billion, and the bottom 40% approximately $29.7 billion, illustrating the skewed income distribution where the majority of income is concentrated among the top earners.

Assessing Poverty Levels

Brazil’s population is approximately 150 million people. Assume each household makes the average income within its quintile, and the poverty line is set at $400 per capita. To estimate the number of individuals living below this threshold, we need to analyze the income distribution further.

Since income data is grouped into quintiles, and the bottom 20% earn only 3% of total income, it suggests a significant portion of this population likely falls below the poverty line, especially considering that many households within this quintile will make less than $400 per capita. If we suppose, conservatively, that all households in the bottom quintile are below the poverty line, then:

Number of people in bottom 20% = 150 million × 20% = 30 million.

Given the total income of $9 billion for this group, the average income per person in the bottom quintile is:

$9 billion / 30 million ≈ $300 per capita.

Since this is below the $400 poverty threshold, we infer that at least those 30 million individuals are living in poverty. The remaining 120 million in the population are less likely to be in poverty, assuming a stratified income distribution within higher quintiles.

Impacts of Income Transfer on Inequality and Poverty

Suppose a transfer of 1% of total national income ($300 billion × 1% = $3 billion) is made from the richest 20% to the poorest 20%. This redistribution would slightly narrow the income gap, reducing inequality, which can be visualized by a Lorenz curve moving closer to the line of equality. As the transfer increases the income of the bottom 20%, the relative position of the Lorenz curve shifts, decreasing the Gini coefficient.

Concerning poverty, this transfer would improve income levels at the lower end. If the entire transfer is distributed evenly among the 30 million individuals in the bottom quintile, each person would receive:

$3 billion / 30 million ≈ $100

adding to their income and possibly lifting some above the poverty line. This targeted income transfer would significantly reduce the number of people living in poverty, demonstrating an effective policy instrument to reduce absolute poverty levels and narrow income inequality.

In conclusion, analyzing Brazil’s income distribution through the Lorenz curve, Gini coefficient, and simulated redistribution highlights persistent inequality issues and the potential policy pathways to promote more equitable growth and social inclusion. Data-driven assessments are essential for designing effective development strategies, ensuring that economic benefits reach the broader population and contribute to sustainable development goals.

References

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